As per the doc (https://docs.aws.amazon.com/apigateway/latest/developerguide/rest-api-mutual-tls.html), private APIs don't support mutual TLS.
In order to do that, you would need something like: Client -> NLB -> Proxy (to handle mTLS) -> interface VPC endpoint for API Gateway -> private API
You can have a look at this blog post: https://aws.amazon.com/pt/blogs/aws-brasil/suportando-mutual-tls-mtls-utilizando-certificados-do-icp-brasil-para-o-open-banking-no-brasil-utilizando-o-amazon-api-gateway/, it's in Portuguese but you will see the architecture, and can probably translate.
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